Concerning an analog of the Stolz angle for the unit ball in $\mathbb C^n$

N. A. Shirokov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 25, Tome 247 (1997), pp. 276-297

Résumé

By a $(\rho,c,q)$-wedge in the unit ball $\mathbb B^n\subset\mathbb C^n$ we mean the union of the sets $\mathbb B^n_\rho$ and $E_{c,q}(e_0)$, where $\mathbb B^n_\rho=\{z\in\mathbb C^n:|z|\le\rho\}$, $0<\rho<1$, $|e_0|=1$, $0<q<1$, $\rho>1-\frac{(1-q)^2}{2(1+c^2)}$, \begin{gather*} E_{c,q}(e_0)=\{z\in\mathbb B^n:|\operatorname{Im}(1-(z,e_0))|\le c\operatorname{Re}(1-(z,e_0)); \\ |z|^2-|(z,e_0)|^2\le q(1-|(z,e_0)|^2)\} \end{gather*} ($(z,\xi)$ is the usual scalar product in $\mathbb C^n$). We denote by $T_a$, $a\in\mathbb B^n$, $a\ne0$, the intersection of $\mathbb B^n$ and the hyperplane $\{z:(z,a)=|a|^2\}$. The paper contains a description of the sets $Z$ of the form $\bigcup\limits_{a\in A} T_a$, where $A$ belongs to a finite union of $(\rho,c,q)$-wedges with $0<q<\frac12$ that may occur as zero-sets or interpolation sets for functions belonging to $H^\infty(\mathbb B^n)$.